Evaluate the following integrals:

Question:

Evaluate the following integrals:

$\int \frac{x^{5}+x^{-2}+2}{x^{2}} d x$

Solution:

Given:

$\int \frac{x^{5}+x^{-2}+2}{x^{2}} d x$

By Splitting, we get,

$\Rightarrow \int\left(\frac{x^{5}}{x^{2}}+\frac{x^{-2}}{x^{2}}+\frac{2}{x^{2}}\right) d x$

$\Rightarrow \int\left(x^{5} \times x^{-2}+x^{-2} \times x^{-2}+2 \times x^{-2}\right) d x$

By applying,

$\Rightarrow \int\left(x^{5-2}+x^{-2-2}+2 x^{-2}\right) d x$

$\Rightarrow \int\left(x^{3}+x^{-4}+2 x^{-2}\right) d x$

By Splitting, we get,

$\int x^{n} d x=\frac{x^{n+1}}{n+1}$

$\Rightarrow \frac{x^{3+1}}{3+1}+\frac{x^{-4+1}}{-4+1}+\frac{2 x^{-2+1}}{-2+1}+c$

$\Rightarrow \frac{x^{4}}{4}+\frac{x^{-3}}{-3}+\frac{2 x^{-1}}{-1}+c$

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